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arXiv · 2205.02159

Sobolev Differentiability Properties of Logarithmic Modulus of Real Analytic Functions

Ziming Shi·Ruixiang Zhang
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Abstract

Let $f$ be the germ of a real analytic function at the origin in $\mathbb{R}^n $ for $n \geq 2$, and suppose the codimension of the zero set of $f$ at $\mathbf{0}$ is at least $2$. We show that $\log |f|$ is $W^{1,1}_{\operatorname{loc}}$ near $\mathbf{0}$. In particular, this implies the differential inequality $|\nabla f |\leq V |f|$ holds with $V \in L^1_{\operatorname{loc}}$.

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Ziming Shi, Ruixiang Zhang. 2025-11-10. Sobolev Differentiability Properties of Logarithmic Modulus of Real Analytic Functions. https://arxiv.org/abs/2205.02159

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